In this section, we investigate the concentration of the three M-estimators taken as example in this article in a corrupted, heavy-tailed setting. The goal will be to recover deviations similar to the one we would have in a Gaussian setting, but when the data are not Gaussian. The gold standard in this context is the deviation of the empirical mean in a Gaussian setting (see [1]}). If \(X_1,\dots ,X_n\)  are i.i.d from \(\mathcal {N}(\mu ,\sigma ^2)\)  for some \(\mu \in \mathbb {R}\)  and \(\sigma >0\) , then for all \(t>0\) ,
\(\mathbb {P}\left(\left|\frac{1}{n}\sum _{i=1}^n X_i-\mu \right|>\sigma \sqrt{\frac{t}{2n}} \right)\le e^{-t}.\) 
